A Higher-order Calculus and Theory Abstraction

Zhaohui Luo

Abstract: We present a higher-order calculus Sigma
CC_\subset that can be seen as an extension of
Coquand-Huet's calculus of constructions (CC) [CH88] by adding
strong sum types and including propositions as types. Strong sum
types in Sigma CC_\subset provides a useful module mechanism
so that abstract structures can be naturally expressed and theories
can be thoroughly abstracted, leading to a comprehensive
structuring of mathematical texts in proof development. Including
propositions as types solves the technical difficulty that adding
(type-indexed) strong sums to the proposition level of CC results
in inconsistency [Coq86a].

We give an omega-Set model construction of Sigma
CC_\subset using the idea developed by E. Moggi
[Mog85][LM88], which entails the consistency of the calculus and
establishes the theoretical soundness of using strong sum types to
express abstract structures and structurally develop theories in
the calculus.

Extending Sigma CC_\subset with an infinite type
hierarchy results in the system Sigma CC \stackrel{\infty}
{\subset}, which is more expressive and more flexible for
structuring theories. Sigma CC \stackrel{\infty}{\subset} is
still strongly normalizing and henceforth consistent, although its
type levels are weakly impredicative.